Note: This is a working paper which will be expanded/updated frequently. All suggestions for improvement are welcome. The directory gifi.stat.ucla.edu/rank has a pdf version, the complete Rmd file with all code chunks, the bib file, and the R source code.

1 Introduction

We have a symmetric matric \(B\) that we want to approximate, in the least squares sense, by a symmetric positive semi-definite matrix of rank less than or equal to \(p\). This is a problem that arises in various forms of multivariate analysis, specifically in factor analysis and multidimensinal scaling. A brief history is in De Leeuw (1974).

Thus the problem is to minimize \[\begin{equation}\label{E:stress} \sigma(X):=\mathbf{SSQ}\ (B-XX') \end{equation}\] over \(X\in\mathbb{R}^{n\times p}\). The stationary equations are \[\begin{equation}\label{E:steq} BX=X(X'X). \end{equation}\]

Clearly the loss function is in variant under rotation, which means we can require without loss of generality that \(X'X\) is diagonal. Thus each column of the optimal \(X\) is either zero or an eigenvector corresponding with a positive eigenvalue of \(B\). At stationary points \(\sigma(X)=\mathbf{SSQ}(B)-\mathbf{tr}\ X'BX\). If the signature of \(B\) is \((n_+,n_-,n_0)\) then the optimal \(X\) has \(\min(n_+,p)\) non-zero columns equal to \(\sqrt{\lambda_s}k_s\), where the \(k_s\) are normalized eigenvectors corresponding to positive eigenvalues \(\lambda_s\), and \(\max(0,p-n_+)\) zero columns. Although the optimal \(X\) is never unique, the optimal \(XX'\) is unique if and only if \(\lambda_p>\lambda_{p+1}\).

If \(B=K\Lambda K'\) is a complete eigen-decomposition, with eigenvalues in decreasing order along the diagonal, then \[\begin{equation}\label{E:decomp} \Gamma_p(B)=K_p\Lambda_p^\frac12 \end{equation}\]

where the \(k_s\) are the normalized eigenvectors corresponding with the \(p\) largest eigenvalues. To guarantee differentiability we assume throughout that \(\lambda_p>\lambda_{p+1}\) as well as \(\lambda_p>0\).

2 Derivative

2.1 Formula

Theorem 1 gives an expression for the first order term in the Taylor expansion \[\begin{equation}\label{E:taylor} \Gamma_p(B+E)=\Gamma_p(B)+\mathcal{D}\Gamma_p(B)(E)+o(\|E\|). \end{equation}\]

Note that \(\Gamma_p\) is defined on the space of real doubly-centered symmetric matrices \(\mathcal{S}^{n\times n}\), with values in that same space. The derivative \(\mathcal{D}\Gamma_p(B)\) at \(B\) is a linear operator from \(\mathcal{S}^{n\times n}\) to \(\mathcal{S}^{n\times n}\), which we apply to the real symmetric and doubly-centered perturbation \(E\). The symmetry makes our results different from those of De Leeuw (2008). Define the matrix \(\Xi:=K'EK\) with elements \(\xi_{st}\).

Theorem 1: [Projection Derivative] \[\begin{equation}\label{E:gamder} \mathcal{D}\Gamma_p(B)(E)=\sum_{s=1}^p\xi_{ss}k_sk_s' -\sum_{s=1}^p\sum_{\substack{t=1\\ t\not=s}}^n\frac{\lambda_s}{\lambda_t-\lambda_s}\xi_{st}(k_tk_s'+k_sk_t'). \end{equation}\] Proof: From De Leeuw (2007) (and many other places) we know that \[\begin{align*} \lambda_s(B+E)&=\lambda_s+\xi_{ss}+o(\|E\|),\\ k_s(B+E)&=k_s-(B-\lambda_sI)^+Ek_s+o(\|E\|). \end{align*}\] Thus \[\begin{align*} \Gamma_p(B+E)&=\sum_{s=1}^p (\lambda_s+\xi_{ss})(k_s-(B-\lambda_sI)^+Ek_s)(k_s-(B-\lambda_sI)^+Ek_s)'+o(\|E\|)=\\ &=\Gamma_p(B)+\sum_{s=1}^p\left\{\xi_{ss}k_sk_s'-\lambda_sk_sk_s'E(B-\lambda_sI)^+-\lambda_s(B-\lambda_sI)^+Ek_sk_s'\right\}+o(\|E\|)=\\ &=\Gamma_p(B)+\sum_{s=1}^p\left\{\xi_{ss}k_sk_s'-\sum_{\substack{t=1\\ t\not=s}}^n\frac{\lambda_s}{\lambda_t-\lambda_s}\xi_{st}(k_sk_t'+k_tk_s')\right\}+o(\|E\|). \end{align*}\]

QED

Note that if \(B\) and \(E\) are SDC, then so is \(\mathcal{D}\Gamma_p(B)(E)\).

We generate some random data to illustrate theorem 1.

set.seed(12345)
eps <- .001
b <- crossprod (matrix (rnorm(400), 100, 4)) / 100
e <- eps * crossprod (matrix (rnorm(400), 100, 4)) / 100

Then compute \(\Gamma_p(B + E)\).

##      [,1]            [,2]            [,3]            [,4]           
## [1,]    1.2781679170    0.1413226188    0.0155429929    0.2557498035
## [2,]    0.1413226188    0.3754844674   -0.3835591107    0.4570154983
## [3,]    0.0155429929   -0.3835591107    0.4126808466   -0.4559121477
## [4,]    0.2557498035    0.4570154983   -0.4559121477    0.5619744567

Then compute the zero-order approximation \(\Gamma_p(B)\).

##      [,1]            [,2]            [,3]            [,4]           
## [1,]    1.2773620868    0.1408232165    0.0156044480    0.2562189505
## [2,]    0.1408232165    0.3753719549   -0.3833537174    0.4567557210
## [3,]    0.0156044480   -0.3833537174    0.4122604048   -0.4554195169
## [4,]    0.2562189505    0.4567557210   -0.4554195169    0.5616655283

And then compute the first-order approximation \(\Gamma_p(B)+\mathcal{D}\Gamma_p(B)(E)\), using the program perturb() from the appendix.

##      [,1]            [,2]            [,3]            [,4]           
## [1,]    1.2781680631    0.1413211691    0.0155430195    0.2557508933
## [2,]    0.1413211691    0.3754850527   -0.3835595774    0.4570158930
## [3,]    0.0155430195   -0.3835595774    0.4126805687   -0.4559116839
## [4,]    0.2557508933    0.4570158930   -0.4559116839    0.5619738470

Clearly the first order approximates is better than the zero order one . For zero order the sum of the absolute values of the approximation errors is 0.0056233246, and for first order it is 0.0000094019.

2.2 Structure

To get more insight into the structure of the derivative we look at its eigenvalues and eigenvectors, i.e. those SDC perturbations \(E\) for which \(\mathcal{D}\Gamma_p(B)(E)=\omega E\) for some \(\omega\). We investigate, again following De Leeuw (2008), perturbation matrices \(E\) of the form \(E=k_\alpha k_\beta'+k_\beta k_\alpha'\), with the \(k_\alpha\) the eigenvectors of \(B\). It suffices to look at those \(\frac12 n(n-1)\) perturbations for which \(\alpha<\beta\). Note they are orthogonal and consequenty are a basis for the space of SDC matrices. Also \[\begin{equation}\label{E:kron} \xi_{st}=k_s'Ek_t=k_s'k_\alpha k_\beta'k_t+k_s'k_\beta k_\alpha'k_t=\delta^{s\alpha}\delta^{t\beta}+\delta^{s\beta}\delta^{t\alpha}. \end{equation}\] Theorem 2: [Projection Eigen Structure] For \(\alpha<\beta\) \[\begin{equation}\label{E:struct} \mathcal{D}\Gamma_p(B)(k_\alpha k_\beta'+k_\beta k_\alpha')= \begin{cases} k_\alpha k_\beta'+k_\beta k_\alpha'&\text{ if }1\leq\alpha<\beta\leq p,\\ \frac{\lambda_\alpha}{\lambda_\alpha-\lambda_\beta}(k_\alpha k_\beta'+k_\beta k_\alpha')&\text{ if }1\leq\alpha\leq p<p+1\leq\beta\leq n,\\ 0&\text{ if }p+1\leq\alpha<\beta\leq n. \end{cases} \end{equation}\]

Proof: From \(\eqref{E:kron}\) we have \(\xi_{ss}=0\) if \(\alpha<\beta\). Thus the first term on the rhs of \(\eqref{E:gamder}\) is zero. It remains to evaluate

\[\begin{equation}\label{E:term} \sum_{s=1}^p\sum_{\substack{t=1\\ t\not=s}}^n\frac{\lambda_s}{\lambda_t-\lambda_s}(\delta^{s\alpha}\delta^{t\beta}+\delta^{s\beta}\delta^{t\alpha})(k_sk_t'+k_tk_s'). \end{equation}\]

If \(\alpha>p\) then \(\beta>p\) and thus \(\eqref{E:term}\) is zero. If \(\alpha<=p\) and \(\beta>p\) then \(\delta^{s\beta}=0\) and

\[ \sum_{s=1}^p\sum_{\substack{t=1\\ t\not=s}}^n\frac{\lambda_s}{\lambda_t-\lambda_s}(\delta^{s\alpha}\delta^{t\beta}+\delta^{s\beta}\delta^{t\alpha})(k_sk_t'+k_tk_s')= \frac{\lambda_\alpha}{\lambda_\beta-\lambda_\alpha}(k_\alpha k_\beta'+k_\beta k_\alpha'). \] If \(\alpha<\beta\leq p\) then \[ \sum_{s=1}^p\sum_{\substack{t=1\\ t\not=s}}^n\frac{\lambda_s}{\lambda_t-\lambda_s}(\delta^{s\alpha}\delta^{t\beta}+\delta^{s\beta}\delta^{t\alpha})(k_sk_t'+k_tk_s')= \frac{\lambda_\alpha}{\lambda_\beta-\lambda_\alpha}(k_\alpha k_\beta'+k_\beta k_\alpha')+\frac{\lambda_\beta}{\lambda_\alpha-\lambda_\beta}(k_\alpha k_\beta'+k_\beta k_\alpha')=k_\alpha k_\beta'+k_\beta k_\alpha'. \] QED

Corollary 1: [Projection Eigen Values] The derivative \(\mathcal{D}\Gamma_p(B)\) has \(\frac12 p(p-1)\) eigenvalues equal to \(+1\), \(\frac12 (n-p)(n-p-1)\) eigenvalues equal to zero, and \(p(n-p)\) eigenvalues equal to \(\frac{\lambda_\alpha}{\lambda_\alpha-\lambda_\beta}\), where \(1\leq\alpha<\beta\leq n\) and the \(\lambda_\alpha\) are the ordered eigenvalues of \(B\).

Note that if \(B\) is positive definite then the \(p(n-p)\) eigenvalues for \(\alpha\leq p\) and \(\beta>p\) are larger than one, if \(B\) is positive semi-definite of rank \(p\) they are equal to one.

3 Thomson’s Factor Analysis Algorithm

Thomson (1934) proposed an alternating least squares algorithm to find factor analysis solutions. We minimize the sum of squares \[ \sigma(X,\Delta)=\mathbf{SSQ}(B-\Delta-XX') \] over loadings \(X\) and the diagonal matrix of uniquenesses \(\Delta\). The algorithm starts with \(\Delta^{(0)}\) and alternates \[\begin{align*} C^{(k)}&=\Gamma_p(B-\Delta^{(k)}),\\ \Delta^{(k+1)}&=\mathbf{diag}(B-C^{(k)}). \end{align*}\]

By substituting the first step into the second step we can think of the algorithm as a map \(\tau\) that updates uniquenesses in a vector \(\delta^{(k)}\) to a vector \(\delta^{(k+1)}\). A straightforward application of theorem 1 shows

\[ \mathcal{D}_j\tau_i(\delta)=\sum_{s=1}^p\left\{k_{is}^2k_{js}^2-2\sum_{\substack{t=1\\ t\not=s}}^n\frac{\lambda_s}{\lambda_t-\lambda_s}k_{is}k_{it}k_{js}k_{jt}\right\}. \] using the eigenvalues and eigenvectors of \(B-\Delta\).

As an example we use the Harman.Burt data from the psych package (Revelle (2016)).

data(Harman, package = "psych")
h3 <- thomson (Harman.Burt, p = 3, verbose = FALSE, eps = 1e-15, itmax = 1000)
h2 <- thomson (Harman.Burt, p = 2, verbose = FALSE, eps = 1e-15, itmax = 1000)
h1 <- thomson (Harman.Burt, p = 1, verbose = FALSE, eps = 1e-15, itmax = 1000)

In one dimension we need 21 iterations to get to loss function value 0.5514339263, with an estimated convergence rate of 0.4818706151. In two dimensions 107 iterations get us to loss 0.1486134802 with estimated rate 0.8746643597. And in three dimensions 435 iterations get us to 0.0312374332 and rate 0.970287468. The estimated convergence rates correspond closely to the largest eigenvalues of the Jacobian, computed by the function tderivative.

mprint (eigen (tderivative (Harman.Burt, h1$u, p=1))$values)
## [1]    0.4819142774    0.3684002138    0.2782070726    0.2446414810
## [5]    0.1830500694    0.1244703151    0.1022921260    0.0597542139
mprint (eigen (tderivative (Harman.Burt, h2$u, p=2))$values)
## [1]    0.8746643733    0.6956043771    0.4674777799    0.4166336265
## [5]    0.3547709022    0.2311944490    0.1756869942    0.0967558960
mprint (eigen (tderivative (Harman.Burt, h3$u, p=3))$values)
## [1]    0.9702874840    0.8848642535    0.8207644608    0.6522238451
## [5]    0.4197176966    0.3589311780    0.2335541555    0.1622049624

4 Appendix: Code

4.1 auxilary.R

ei <- function (i, n) {
  return (ifelse(i == (1:n), 1, 0))
}

aij <- function (i, j, n) {
  u <- ei(i, n) - ei (j, n)
  return (outer (u, u))
}

kdelta <- function (i, j) {
  ifelse (i == j, 1 , 0)
}

mprint <- function (x, d = 10, w = 15) {
  print (noquote (formatC (
    x, di = d, wi = w, fo = "f"
  )))
}

mnorm <- function (x) {
  return (sqrt (sum (x ^ 2)))
}

basis <- function (i, j, n) {
  s <- sqrt (2) / 2
  a <- matrix (0, n, n)
  if (i == j)
    a[i, i] <- 1
  else {
    a[i, j] <- a[j, i] <- s
  }
  return (a)
}

doubleCenter <- function (x) {
  n <- nrow (x)
  j <- diag(n) - (1 / n)
  return (j %*% x %*% j)
}

squareDist <- function (x) {
  d <- diag (x)
  return (outer (d, d, "+") - 2 * x)
}

4.2 thomson.R

thomson <-
  function (b,
            uold = rep (0, nrow (b)),
            p = 2,
            itmax = 1000,
            eps = 1e-10,
            verbose = TRUE) {
    bold <- b - diag (uold)
    cold <- lrank (bold, p)
    fold <- sum ((bold - cold) ^ 2)
    eold <- Inf
    itel <- 1
    repeat {
      unew <- diag (b - cold)
      enew <- mnorm (uold - unew)
      rnew <- enew / eold
      bnew <- b - diag (unew)
      cnew <- lrank (bnew, p)
      fnew <- sum ((bnew - cnew) ^ 2)
      if (verbose) {
        cat (
          formatC (itel, width = 4, format = "d"),
          formatC (
            fold,
            digits = 10,
            width = 15,
            format = "f"
          ),
          formatC (
            fnew,
            digits = 10,
            width = 15,
            format = "f"
          ),
          formatC (
            enew,
            digits = 10,
            width = 15,
            format = "f"
          ),
          formatC (
            rnew,
            digits = 10,
            width = 15,
            format = "f"
          ),
          "\n"
        )
      }
      if ((itel == itmax) || (fold - fnew) < eps)
        break
      itel <- itel + 1
      uold <- unew
      fold <- fnew
      cold <- cnew
      bold <- bnew
      eold <- enew
    }
    return (list (
      u = unew,
      f = fnew,
      e = enew,
      r = rnew,
      itel = itel
    ))
  }

tmap <- function (b, u, p = 2) {
  c <- lrank (b - diag (u))
  return (diag (b - c))
}

tderivative <- function (b, u, p = 2) {
  n <- length (u)
  e <- eigen (b - diag (u))
  k <- e$vectors
  l <- e$values
  m <- matrix (0, n, n)
  for (i in 1:n)
    for (j in 1:n)
      for (s in 1:p) {
          m[i,j] <- m[i,j] + (k[i,s] ^ 2) * (k[j, s] ^ 2)
        for (t in 1:n) {
          if (s == t)
            next
          xi <- l[s] / (l[t] - l[s])
          m[i, j] <-
            m[i, j] - 2 * xi * k[i, s] * k[i, t] * k[j, s] * k[j, t]
        }
      }
  return (m)
}

References

De Leeuw, J. 1974. “Approximation of a Real Symmetric Matrix by a Positive Semidefinite Matrix of Rank R.” Technical Memorandum TM-74-1229-10. Murray Hill, N.J.: Bell Telephone Laboratories. http://www.stat.ucla.edu/~deleeuw/janspubs/1974/reports/deleeuw_R_74c.pdf.

———. 2007. “Derivatives of Generalized Eigen Systems with Applications.” Preprint Series 528. Los Angeles, CA: UCLA Department of Statistics. http://www.stat.ucla.edu/~deleeuw/janspubs/2007/reports/deleeuw_R_07c.pdf.

———. 2008. “Derivatives of Fixed-Rank Approximations.” Preprint Series 547. Los Angeles, CA: UCLA Department of Statistics. http://www.stat.ucla.edu/~deleeuw/janspubs/2008/reports/deleeuw_R_08b.pdf.

Revelle, W. 2016. psych: Procedures for Psychological, Psychometric, and Personality Research. Evanston, Illinois: Northwestern University. https://CRAN.R-project.org/package=psych.

Thomson, G.H. 1934. “Hotelling’s Method Modfiied to Give Spearman’s g.” Journal of Educational Psychology 25: 366–74.